Cross section (geometry)

Cross Section (Geometry)

Cross section (geometry)

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In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc. More plainly, when cutting an object into slices one gets many parallel cross-sections.

Cavalieri's principle states that solids with corresponding cross-sections of equal areas have equal volumes.

The cross-sectional area (<math>A'</math>) of an object when viewed from a particular angle is the total area of the orthographic projection of the object from that angle. For example, a cylinder of height h and radius r has <math>A' = pi r^2</math> when viewed along its central axis, and <math>A' = 2 pi rh</math> when viewed from an orthogonal direction. A sphere of radius r has <math>A' = pi r^2</math> when viewed from any angle. More generically, <math>A'</math> can be calculated by evaluating the following surface integral:

<math> A' = iint limits_mathrm dmathbf cdot mathbf, </math>


where <math>mathbf</math> is a unit vector pointing along the viewing direction toward the viewer, <math>dmathbf</math> is a surface element with outward-pointing normal, and the integral is taken only over the top-most surface, that part of the surface that is "visible" from the perspective of the viewer. For a convex body, each ray through the object from the viewer's perspective crosses just two surfaces. For such objects, the...
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